Electronic copy available at: http://ssrn.com/abstract=2579670

Target date funds: Marketing or Finance?

An Chena, Carla Mereub, Robert Stelzerc

aUniversity of Ulm, Institute of Insurance Science, Helmholtzstrasse 20, 89081 Ulm, Germany.bUniversity of Ulm, Institute of Mathematical Finance, Helmholtzstrasse 22, 89081 Ulm,

Germany.cUniversity of Ulm, Institute of Mathematical Finance, Helmholtzstrasse 18, 89081 Ulm,

Germany.

Abstract

Since Target Date Funds (TDFs) became one of the default investment strategies

for the 401(k) defined contribution (DC) beneficiaries, they have developed rapidly.

Usually they are structured according to the principle young people should invest

more in equities. Is this really a good recommendation for DC beneficiaries to

manage their investment risk? The present paper relies on dynamic asset allocation

to investigate how to optimally structure TDFs by realistically modelling the con-

tributions made to 401(k) plans. We show that stochastic contributions can play

an essential role in the determination of optimal investment strategies. Depending

on the correlation of the contribution process with the markets stock, we find that

an age-increasing equity holding can be optimal too. This result highly depends on

how the contribution rule is defined.

Keywords: Utility theory, Optimal asset allocation, Defined contribution, Target

date fund

1. Introduction

Target date funds (TDF) are investment funds with a prespecified maturity (tar-

get date). Because of their structure, these funds place themselves in the category of

life-cycle funds, rather than in the category of life-style funds where the target is

the risk profile of the investor. TDFs have developed very rapidly, particularly after

they became one of the default investment strategies of a 401(k) defined contribution

Email addresses: an.chen@uni-ulm.de (An Chen), carla.mereu@uni-ulm.de(Carla Mereu), robert.stelzer@uni-ulm.de (Robert Stelzer)

Electronic copy available at: http://ssrn.com/abstract=2579670

(DC) beneficiary.1 According to Morningstar Fund Research (2012), assets in the

TDFs have grown from 71 billion US dollars at the end of 2005 to approximately

378 billion dollars at the end of 2011. These funds are directly coupled with the

retirement year of the DC plan investors and have the advantage that the investors

do not have to choose a number of investments, but only a single fund. The main

mechanism behind these TDFs is: those who retire later shall invest more in eq-

uity, while those who retire earlier shall invest less in equity. In other words, equity

holding in TDFs shall decrease in age. Therefore, TDFs are usually identified by

practitioners with glide paths, i.e. the decreasing curve of the equity holding (as

a fraction of wealth) over time.

But is this shaping of target date funds really a good recommendation for DC

beneficiaries to manage the investment risks? Shall every DC beneficiary who re-

tires in 2050 take the same target date fund, independent of his income, and risk

preference? Is the popular financial advice just anecdotal evidence? Or can it be

justified by rigorous theory?

There is few literature aiming to find an optimal equity holding which justifies

the target date fund.2 At first sight, target date funds are inconsistent with Mertons

portfolio (c.f. Merton (1969) and Merton (1971)), which has sometimes been con-

sidered as an economic puzzle. For an investor with a constant relative risk aversion

preference, Mertons optimal portfolio prescribes to invest a constant proportion of

wealth in equity (constant-mix strategy is optimal), i.e. the optimal portfolio does

not depend on time/age. One of the most famous rigorous economic justifications

for the age-dependent (particularly age-decreasing) investment behavior is given in

Jagannathan & Kocherlakota (1996). In their paper, by using economic reasonings,

1DC beneficiaries need to bear the entire investment risks and management of their pensionplans. In the US, they can manage it by so called Individual Retirement Accounts, or morefrequently by making contributions to 401(k) plans, where the amount of contributions mainlydepends on the development of an employees salary (income).

2In the academic literature, the study of target date funds has been mostly based on simulationstudies: taking several prevailing strategies (target date fund and constant mix strategy), comparethem and find the best strategy among them. For instance, Spitzer & Singh (2008) comparethe target date funds with constant-mix strategy (50-50%) by examining the ruin probability viabootstrap simulation and rolling period analysis and show that a constant mix strategy outperformsthe TDF all the time.

2

the robustness of the arguments justifying glide paths is called into question. They

discuss the three prevailing hypotheses supporting it and accept human capital, in

form of present value of expected future earnings, as the only valid reason to solve

this apparent puzzle. Using a simplified model and some qualitative arguments, they

also show the validity of the argument younger people shall invest more in stocks

because younger people have more labor income ahead under certain conditions,

e.g. if the correlation of labor income with stock returns is not too high.

In a DC pension plan, the beneficiary makes contributions whose amount mainly

depends on the development of her salary (income). Based on this fact and also

motivated by Jagannathan & Kocherlakota (1996), we use an optimal dynamic asset

allocation approach in continuous time to investigate the role of labor income in the

form of stochastic contributions in the utility maximization of the wealth of a DC

beneficiary. We find that stochastic contributions can play an essential role in the

determination of the optimal equity holding. More interestingly, it depends much on

how the contribution rule is set. We mainly discuss two models as representatives

of the following general forms:

1) Contributions are adjusted as a varying proportion of the fund value, where the

proportion is allowed to be stochastic but just depends on the salary process.

2) Contributions depend only on the salary process.

We show that in the first case, correlation between the asset and salary risk plays the

dominant role in the asset allocation, which decides whether the equity proportion

is equal to, higher or lower than Mertons portfolio. The effect of human capital,

interpreted as discounted value of future wages, is rather secondary and influences

the magnitude of the equity holding when there is some correlation between the two

risks.

In the second case, the effect of human capital becomes more relevant. The

resulting equity holding is a time-dependent and, in most -but not all- cases, time-

decreasing proportion of wealth which suggests a higher equity holding than in Mer-

tons portfolio. We will see that the correlation between the asset and salary risk is

still a deciding factor in the equity holding. However, uncorrelated risks do not im-

ply an equity holding identical to Mertons portfolio. Through our analysis, we show

3

that the optimality of glide paths (age-decreasing equity holding) can be in most

cases verified in the presence of stochastic contributions. However, in some extreme

cases (e.g. the asset and income risks are highly positively correlated), the optimal

equity holding could recommend that an older beneficiary shall invest more in stocks

than a younger one. In other words, it is not necessarily true that one should use

a glide path as reference, as already conjectured in Jagannathan & Kocherlakota

(1996). Let us mention here that similar conclusions are also drawn by Dybvig &

Liu (2010). However, the main focus in their paper is on the impact of retirement

flexibility and borrowing constraints in determining the optimal consumption and

investment for the problem of maximizing the utility from consumption and be-

quest. Our model neglects on the one side the fact that the retirement time could

be random and the optimal stopping problem deriving from that, on the other side

better fits the structure of TDFs. Moreover, from a technical point of view, Dybvig

& Liu (2010) are able to reduce the optimization problem to a time-independent

ODE, whereas in our case of maximizing the utility of terminal wealth complex,

non-linear parabolic equations need to be considered.

The mathematical foundation of the current paper is optimal dynamic asset al-

location (in an incomplete financial market). There exists a stream of literature

on optimal dynamic asset allocation applied for a DC pension scheme with diverse

financial market settings and different preferences.3 In the present paper, we incor-

porate untradable salary risk in the contribution process and analyze the optimal

asset allocation for a target date fund in the context of defined contribution schemes.

3Gao (2008) studies this problem under stochastic interest rates. Boulier, Huang & Taillard(2001) solve it under the constraint that a guaranteed amount is provided to the beneficiary ina stochastic interest rate framework. Blake, Wright & Zhang (2013) investigate this problemwith a loss-averse preference (instead of using a conventional utility function) and study so calledtarget-driven investing. Incorporating salary (income) risk by modelling it as a geometric Brown-ian motion, Zhang, Korn & Ewald (2007) focus on the influence of inflation on pension products.Di Giacinto, Federico, Gozzi & Vigna (2014) emphasize the possibility that the retirees can post-pone annuity purchasing after retirement, i.e. they are provided with an income drawdown option.Cairns, Blake & Dowd (2006) consider this optimization problem under a utility function whichuses the plan members salary as a numeraire. Our formulation differs from Cairns, Blake & Dowd(2006): In Cairns, Blake & Dowd (2006), the pension beneficiary maximizes the expected utility ofthe terminal wealth divided by his terminal salary, while in our paper we maximize the expectedutility of the terminal wealth. This objective is the traditional one considered in the literature,and appears to us to be more natural.

4

We rely on dynamic programming and use the dimension-reduction technique devel-

oped in Chen, Mereu & Stelzer (2014) to solve the optimal asset allocation problem.

The remainder of the paper is organized as follows. Section 2 describes the model

setup, introduces our optimization problem and particularly the two contribution

rules. In the subsequent Section 3, we rely on the separation theorem and solve

the optimization problem for the first case in which the contribution is a proportion

of the fund value. We also provide an example of a model with mean-reverting

contributions depending both on the wealth process and the salary, and obtain an

analytical solution. In Section 4, we treat the second case in which the contribution

is a function of salary exclusively and use Chen, Mereu & Stelzer (2014) to solve the

optimization problem. Moreover, we study numerically the effects of correlation and

human capital on the optimal equity holding. Finally, we provide some concluding

remarks in Section 5, and exhibit a set of detailed mathematical derivations in the

appendix.

2. Model setup

On a fixed filtered probability space (,F , {Ft}t[0,T ],P), satisfying the usualhypotheses, consider a financial market consisting of a riskless asset and a risky

asset. From now on let T be a fixed finite time point, representing the age of

retirement4.

(S0, S1) will denote respectively the savings account (riskless asset) and the risky

asset, and we assume that the two assets follow a Black-Scholes model:

dS0t = rS0t dt,

dS1t = S1t dt+ S

1t dW

1t ,

where , r R, > 0 and W 1 is a Brownian motion on the above mentionedfiltered probability space.

The contributions of DC plan investors are usually coupled to their salary.

We assume that the salary process I has stochastic dynamics driven by another

4We neglect the mortality risk, assume that the retirement time is deterministic and focus onthe stochastic contribution risk.

5

Brownian motion W I on the same space, correlated with W 1 with a correlation

coefficient . So, there is a Brownian motion W 2, independent of W 1 such that

W I = W 1 +

1 2W 2.The process It is not constant, but rather describes a time-varying salary (e.g. over-

time payments, bonuses etc.):

dIt = I(t, It) dt+ I(t, It) dWIt , (1)

where I : [0, T ] R+ R, I : [0, T ] R+ R+ are cadlag in time, and locallyLipschitz continuous of at most linear growth in the second variable. We assume

that the fund manager invests at any time t a proportion t of the wealth in the stock

S1 and 1t in the bond S0 with interest rate r, and that the corresponding wealthprocess evolves according to a stochastic process {At }t[0,t], whose dynamics will bespecified in detail later on. In addition, there are contributions continuously flowing

to the plan members individual account at rate ct := f(At , It). The contribution

rule is a function f : R2 R+, f C2, depending on the funds value At and onthe salary process It. There are several special cases, among the ones that can be

considered:

f(At , It) = (It) At , where C2(R). Contributions are here adjusted asa varying proportion of the fund value, where the proportion is allowed to be

stochastic but just depends on the salary process.

f(At , It) = f(It), where f C2 is strictly positive and just depends on thesecond variable. Here the contributions are based on the wage.

We will see that the first case can be treated sometimes in an analytic way (see

Appendix A), while the second one requires more efforts, but reflects more realistic

contribution processes in the DC plan. For instance, f(At , It) = It is the con-

tribution rule most frequently used in practice, i.e. the contribution ct is taken as

a constant fraction (0, 1) of the salary process. This is just a special case off(At , It) = f(It).

Remark 2.1. Under the above assumptions, the contributions ct := f(At , It) are also

Ito diffusions, and will have stochastic dynamics of the type:{dct = C(t, A

t , It, t) dt+

1C(t, A

t , It, t) dW

1t +

2C(t, A

t , It, t) dW

2t ,

c0 = y,(2)

6

where C : [0, T ]R+R+A R and iC : [0, T ]R+R+A R+, i = 1, 2,are deterministic functions obtained in terms of the first and second derivatives off , and A R is a set the strategies take values in. We will therefore sometimesgive directly the contribution process as a general Ito diffusion without referring tothe income dynamics.

We will often talk about a strategy meaning by this the pair (1 , ).Given an investment strategy , the pension wealth process obtained by trading

according to the strategy , {At }t[0,T ], has the following dynamics:

dAt =At tS1t

dS1t +At (1 t)

S0tdS0t + ct dt, (3)

and hence, defining := r, for an initial wealth x R+ we get{

dAt = [At (t + r) + ct] dt+ A

t t dW

1t ,

A0 = x.(4)

Note that this definition reflects the fact that the only allowed additional cash

injections to the fund are due to the continuous payments at rate ct = f(At , It).

A progressively measurable process is said to be admissible, if it takes values in a

fixed convex subset A of R such that T

0|s|2 ds < a.s. and At > 0 for every

t [0, T ].5 We denote by A the set of all admissible strategies.For simplicity, let us assume that A is compact.

Remark 2.2. In the definition of admissibility, one has to ensure that the fund pro-cess never becomes negative. The general contribution process does not necessarilyensure positivity. In our second case in which f(At , It) = f(It) (with f strictlypositive), the wealth process stays positive without any extra requirements on theadmissible strategies.

Recall that the driving Brownian motion W I represents the uncertainty in the

salary, and is spanned by two Brownian motions. One of these, W 2, is assumed not

to be traded in the market. This makes the market incomplete.

Assume that the DC plan investor gets a lump-sum payment at time of retirement

T 6 and wants to maximize the expected utility from this terminal wealth. More

5The square-integrability condition ensures the existence and uniqueness of a solution for Equa-tion (4), provided some regularity assumptions on the contribution rule (e.g. Lipschitz continuityin the first variable) hold, together with some assumptions on the contribution process.

6Most DC pension plans pay out a lump-sum instead of annuities.

7

precisely, we are looking for an optimal investment strategy such that

E[U(A

T )]

= supA

E [U(AT )] ,

where U : R R+ is a CRRA power utility function and A is the set of admissiblestrategies, i.e. for a risk aversion parameter < 1, 6= 0,

U(x) =x

.

The power utility is abundantly used in both theoretical and empirical research be-

cause of its nice analytical tractability. Most importantly, the use of the power utility

is also motivated economically, since the long-run behavior of the economy suggests

that the long run risk aversion cannot strongly depend on wealth, see Campbell &

Viceira (2002).

The value function of the utility maximization problem is given by

v(t, x, y) := supA

E[U(A,t,x,yT )

], (t, x, y) [0, T ) (0,+) (0,+), (5)

where we are taking as controlled process the pair X = (A, I), and the notation

A,t,x,y stands for the first coordinate of the process X starting from the point

(x, y), respectively the initial wealth and the initial income, at time t.

Please note that the process I actually does not depend on the control . Applying

well-known results in stochastic control, see e.g. Pham (2009) Chapter 3, in partic-

ular Theorem 3.5.2, we can write down the HJB equation for the value function of

our control problem:

vt(t, x, y) = supA

{[x( + r) + f(x, y)] vx(t, x, y) + I(t, y)vy(t, x, y)

+1

2(x)2vxx(t, x, y) +

1

2I(t, y)

2vyy(t, x, y) + I(t, y)xvxy(t, x, y)},

(6)

v(T, x, y) =U(x), (x, y).

Here and in the remainder, we will denote the partial derivatives by subscripts, and

often omit the argument of the function in the equations.

8

The solution to the HJB PDE (6) depends much on the contribution rule and

cannot be solved analytically in general. In what follows, we discuss a representative

model of each of the two contribution rules: f(At , It) = (It) At in Section 3 andf(At , It) = f(It) in Section 4.

3. Contribution as a fraction of the fund value

If we are ready to constrain the contribution rule to the form f(x, y) = (y)x,

as a fraction of the fund value, where is exclusively a function of y, in some cases

we are able to achieve analytic solutions.

Example 3.1. Take as contribution rule f(x, y) = (y)x, and assume that the process(It) evolves according to an Ornstein-Uhlenbeck process

7.We assume directly that the dynamics of the process t = (It) are given as

8:{dt = ( t) dt+ dW It ,0 = y,

(7)

where the constant parameters are such that > 0 denotes the mean reversionspeed, and the mean reversion level, whereas > 0 is the volatility. For thismodel, it can be seen (see Appendix A for more details) that the optimal strategyis given by

=(t) =

(

(e(Tt) 1

)) 1(1 )

=

(1 ) 1(1 )

(e(Tt) 1)

. (8)

Let us comment on the optimal strategy (8).

First, it consists of two terms: the first term is the famous Merton portfo-

lio, and the second term is an additional component which accounts for the

(hedgeable) correlated contribution risk. This clear-cut decomposition of the

strategy, particularly filtering out the Merton portfolio, is only possible be-

cause the contribution rule f(x, y) = (y)x allows us to use the separation

7This function is obviously Lipschitz continuous in x, and since the Ornstein-Uhlenbeck processis predictable, there is a solution to Equation (4).

8In the case where the dynamics of the process It are given instead of the dynamics of thecontribution process, up to an application of Itos formula to the process (It), we get a similarresult.

9

0 5 10 15 20 25 302

1.5

1

0.5

0

0.5

1

1.5

2

2.5

t

*

Optimal equity proportion

Optimal proportion, = 0.5

Optimal proportion, = 0

Optimal proportion, = 0.5

Merton ratio

Figure 1: Optimal equity proportion against time in the model of Section 3Parameters: = 0.2, = 0.06, r = 0.02, = 0.05, = 0.02, = 2, T = 30.

ansatz v(t, x, y) = (t, y)U(x) on the value function. The explicit expression

of (t, y) can be found in Appendix A

Second, the correlation coefficient between the tradeable and the untradeable

risk plays the most important role in the magnitude of the strategy. The sign

of decides whether the optimal investment is identical to, higher or lower

than the Merton portfolio. For < 0 (a relative risk aversion larger than 1), a

positive correlation leads to an optimal equity holding lower than the Merton

portfolio, and a negative correlation results in an optimal equity holding higher

than the Merton portfolio. For 0 < < 1 (a relative risk aversion between

0 and 1), the effect of is reversed. A higher correlation means that more

untradable risks can be eliminated through going short in the tradable stock.

Third, the optimal strategy varies in time and converges towards the Merton

portfolio when the time approaches maturity, as can be read from Equation

(8) and is illustrated in Figure 1.

Fourth, compared to the influence of the correlation coefficient , the effect of

income risk (contribution) is secondary. For instance, for = 0, the resulting

optimal strategy coincides with the Merton portfolio: the parameters and

which drive the contribution process do not influence this result. You might

argue that the effect of the income is secondary just due to our modelling, i.e.

10

since we use Ornstein-Uhlenbeck process to model (It), which means that

the contribution might become negative. Therefore, in this place, let us briefly

mention another simpler example which helps us see the secondary effect of the

contribution on the optimal investment strategy: (It) = , i.e. f(x, y) = x,

> 0. In this case, we have a continuous stream of positive contributions

flowing to the pension fund, but the optimal strategy is not time-dependent

and it is easy to see that it coincides with Mertons portfolio. In this simple

example, although the contributions are always strictly positive, the income

has no effect at all.

The analysis in this section shows that adding a contribution (depending on income)

in the form f(At , It) = (It) At to the original Merton setting might lead to a time-dependent optimal portfolio, depending on the specification of (It). Under certain

circumstances, it is also possible to achieve a strategy which decreases in time,

suggesting that younger participants shall optimally hold more equity. However,

in this case, the contribution/income does not play the most important role. The

correlation together with the risk aversion level and income parameter determines

consequently the magnitude of the equity holding.

4. Contribution depending only on the income

In the current section, we look at the more realistic contribution process: f(At , It) =

f(It). The optimization problem becomes much more complicated. No separation

ansatz seems to be adoptable to derive the value function and the optimal strategy,

which has the consequence that explicitly filtering out the Merton portfolio is im-

possible. Some natural questions need to be answered in the next section: will the

income (contribution) become now the dominant effect in the optimal strategy? In

this case, do we always achieve an optimal strategy which decreases in time? Here,

we model the contribution process as a process depending only on the wage, i.e.

f(x, y) = f(y). If f C2(R) is an invertible function, the contribution process itselfwill be again an Ito-diffusion, whose dynamics can be derived from the dynamics

of the wage process I. Slightly changing the setting with respect to the previous

sections, and with a little abuse of notation, we will therefore directly model the

contribution process.

11

In the following, we assume that the contributions have stochastic dynamics given

as: {dct = ct

(C(t) dt+ C(t) dW

It

),

c0 = y,(9)

where C : [0, T ] R and C : [0, T ] R+ are cadlag in time, and y > 0 is theinitial contribution.

Unfortunately, it is very hard to make an educated guess on the solution of the

PDE corresponding to this case, and so we are unable to apply classical verification

methods. To solve this optimal asset allocation problem, Chen, Mereu & Stelzer

(2014) reduce the HJB equation by one dimension to make the optimization problem

well solvable through numerical methods. Chen, Mereu & Stelzer (2014) show that

the value function reads v(t, x, y) = yu(t, xy

), where u is a solution of a reduced

PDE (see Appendix B for more details and a precise statement of the theorem).

They also show that the optimal strategy is described by t = h(t, At , ct

)where

h(t, x, y) :=

uzz(t,xy

)

uz(t,xy

)xy

C(t)

1 uzz(t,xy )uz(t,

xy

)xy

1

, if it belongs to A a.e.(10)

This also proves that the optimal strategy will depend on the current wealth x and

income y only though their ratio z = xy.

Note that the strategy in (10) also consists of two parts:

The first term has a similar form as, but is not identical to, the Merton port-

folio. In fact, it is now impossible to explicitly obtain the Merton portfolio as

a separate summand. The difference lies in the coefficient in the denominator.

Since the separation ansatz does not work in this case, the additional contri-

bution/income influences also the relative risk aversion of the indirect utility:

We now have a coefficient changing with time and the ratio of wealth over

income, z. This varying RRA is given by uzzuz

z, which does not necessarily

equal 1 . In Chen, Mereu & Stelzer (2014), the authors show that whenz goes to infinity (i.e. the income is negligible compared to the wealth), this

quantity converges to 1 . In other words, asymptotically we are back toMertons case. Due to this effect of the contribution, even when there is no

correlation between asset and contribution risk, we are already obtaining a

time-dependent equity holding.

12

The second term is again the hedging component which accounts for the

(hedgeable) correlated untradable contribution risk. It disappears when = 0

or the relative risk aversion of the indirect utility uzzuz

z converges to 1 ,for z . In this case, since the comparison between the two magnitudes(uzzuz

z and 1 ) depends on specific parameter choices, we can identify theunambiguous impact of .

Following the above two observations, we notice that there are two main factors

driving the choice of the investor:

1. the correlation between the random endowment and the risky asset, which allows

to hedge partially away risks from the random future contribution by choosing

the investment strategy accordingly;

2. the presence of a strictly positive random endowment, which corresponds to the

availability of future incomes, and corresponds to an extra - non financial - wealth,

called human capital.

In the following two subsections, we will analyze these two effects in detail.

4.1. The impact of correlation

With the contribution rule studied in Section 3, the effect of correlation is very

important and the value of the correlation for the qualitative change of the strategy

is = 0. However, in this second contribution rule = 0 is not crucial anymore.

Numerical studies seem to suggest that also in the second contribution rule there

is a value of the correlation determining whether the optimal proportion lies above

or below the Merton ratio, but that this critical value will be now always strictly

bigger than 0 (see later Figure 2). The following result aims at understanding this

issue a little deeper.

Proposition 4.1. In the model of Section 4, assume that C is constant over timeand > r. If the parameters are such that 1 r

C, then for the value

= r

C(1 )> 0 (11)

the corresponding optimal strategy is constant and coincides with the Merton ratio,namely

h(t, x, y) r2(1 )

,

13

if it belongs to A .

Proof. To show the claim, we look for conditions on the correlation such that theoptimal policy in (10) coincides with the Merton ratio. To this end, define for z = x

y

(t, z) := uzz(t, z)uz(t, z)

z (1 ).

We basically have to look for a such that

r2(1 )

= r

2(1 + ) C

(1

1 + 1).

Rearranging the terms, this holds if and only if

r

(1

1 1

1 +

)= C(1 )

(1

1 + 1

1

).

One can then see that this equation is satisfied for = rC(1)

.

That is strictly positive follows from Equation (11) since > r. Moreover, isa correlation coefficient thanks to the assumptions on the parameters.

Remark 4.2. Note that also for (t, z) = 0 (t, z) we would have that the optimalproportion corresponds to the Merton ratio. It seems however hard to prove whetherand for which parameters this holds.From the upcoming numerical results, we can actually conjecture that the value

in Equation (11) is critical in the following sense:

if > , then < r2(1) for all t [0, T )

if < , then > r2(1) for all t [0, T ).

Moreover, for 1 rC

, i.e. for a sufficiently small relative risk aversion parameter(sufficiently big ), all values of the correlation should yield optimal strategies lyingabove the Merton ratio.

Figure 2 plots the optimal equity proportion as a function of time for 5 dif-

ferent values of . For the given parameters, the critical correlation coefficient is

0.1923. Therefore, for all < 0.1923 (i.e. here = 0.95, 0.5, 0), theoptimal equity holding is higher than than the Merton portfolio. Furthermore, it

demonstrates a glide path, a time-decreasing equity-holding. On the contrary, for

all > 0.1923 (i.e. here = 0.5, 0.95), we observe a time-increasing equity holding

which is overall lower than the Merton portfolio. This in particular shows that in

this model the optimal strategy is not necessarily given as a glide path, but can be

14

Figure 2: Optimal equity proportion in the model of Section 4 for different valuesof the correlation parameter .Parameters: T = 30 years, and = 0.4, = 0.04, C = 0.13, C = 0.02, r = 0.02, = 1, z = 15.

0 5 10 15 20 25 300.4

0.2

0

0.2

0.4

0.6

0.8

1

t

*

Optimal equity proportion, z= 15

= 0.95

= 0.5

= 0

= 0.5

= 0.95

Merton ratio

also increasing in time in some cases, depending on the correlation of the contribu-

tion process with the market assets.

If we compare now Figure 2 and Figure 3 for = 1 and = 0.5 respectively(i.e. RRA = 2 and 0.5, respectively), the typical glide path structure of TDFs can

be better justified for less risk-averse agents, because there is a bigger region of s

where a time-decreasing equity holding results as the optimal solution ( 0.1923in the case of Figure 2 and 0.7692 in the case of Figure 3). This is a directconsequence of Proposition 4.1, since the critical value of the correlation for which

the optimal strategy concides with the Merton ratio is decreasing in the RRA (cf.

Equation (11)).

Also, the volatility of the stock plays a fundamental role: for a stock with a higher

volatility, we observe a conservative behavior for a risk-averse investor (cf. Figure

4) already for smaller values of the correlation.

4.2. Human capital

Let us now briefly analyze the effect of human capital on the investment choices.

Usually, human capital is understood as discounted value of future wages, social

security, and other benefits. The total wealth of a person will then include both his

current financial assets and his human capital.

15

Figure 3: Optimal equity proportion in the model of Section 4 for different valuesof the correlation parameter and positive .Parameters: T = 30 years, and = 0.4, = 0.04, C = 0.13, C = 0.02, r = 0.02, = 0.5, z = 15.

0 5 10 15 20 25 300.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

*

Optimal equity proportion, z=15, =0.5

= 0.95

= 0.5

= 0

= 0.5

= 0.95

Merton ratio

In the following, we compare the behavior of a young and an old investor with

the same (relative) risk aversion and the same working category (in the sense that

their salary has the same correlation with the market assets). We distinguish be-

tween the two investors by choosing a different initial value z = x/y and a different

contract duration T . A higher z means a higher initial capital compared to the

initial contribution. The young investor still needs to work for another 30 years

until retirement and starts with a lower initial z = 5, while the old investor only

needs to work for another 10 years and has an initial z = 20. We can consider the

scenario where the young investor chooses the Target Date Fund 2045 and the old

one the Target Date Fund 2025. The resulting optimal equity holdings are plotted

in Figures 4 and 5 respectively.

Comparing the two blue curves (corresponding to = 0.2) in these two graph-

ics, we obtain some expected observations: Young investors have a longer time to

work and have not had much time to accumulate wealth. The ability to work (hu-

man capital) is therefore their largest asset. Older investors have already converted

most of their human capital to financial capital. In this sense, young investors can

borrow from their future income to invest more in the risky asset, which leads to

16

Figure 4: Optimal equity proportion in the model of Section 4 for a young investor.Here z=5, T=30 years, = 0.25 and = 0.2, 0.4, = 0.04, C = 0.13, C = 0.02,r = 0.02, = 1.

0 5 10 15 20 25 300.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t

*

Optimal equity proportion for a young investor, z=5

Optimal proportion, =0.2

Merton ratio, =0.2

Merton ratio, =0.4

Optimal proportion, =0.4

Figure 5: Optimal equity proportion in the model of Section 4 for an old investor.Here z=20, T=10 years, = 0.25 and = 0.2, 0.4, = 0.04, C = 0.13, C = 0.02,r = 0.02, = 1.

0 1 2 3 4 5 6 7 8 9 100.05

0.1

0.15

0.2

0.25

0.3

t

*

Optimal equity proportion for an old investor, z=20

Optimal proportion, =0.2

Merton ratio, =0.2

Merton ratio, =0.4

Optimal proportion, =0.4

a higher equity holding. Sometimes there is another interpretation to human cap-

ital: Young and old investors with the same relative risk aversion want to achieve

the same ideal portfolio (in our case a certain mixture of the risky and risk free

asset). Since the ability to work is not affected by market risk, human capital is

frequently considered more like the risk free asset. Young investors, which have a

17

lot of human capital (i.e. risk free asset), need to invest more in equity to achieve

this ideal portfolio. On the other side, old investors have little human capital and

will consequently invest less in equity to achieve this ideal portfolio.

Note that the above-mentioned expected result can be only achieved for some

specific choices of the parameters. In many situations, the validity of the argument

will be violated, e.g. by choosing a higher correlation coefficient between the market

and income risk or a different stock volatility. In our example, we exhibit how

the investment behavior of the young and old investor will change when we move

from = 0.2 to = 0.4. The resulting optimal equity holding for the young and

old investor for the more volatile risky asset are shown by the dotted curves in

Figures 4 and 5. Both investors investment behavior does not indicate a glide path

and suggests a lower equity holding than the Merton portfolio. In particular, the

young investor rich in human capital shall optimally invest less in the risky asset

than the older one. This example also shows that the optimal equity holding can

demonstrate a rising glide path (dotted curves corresponding to = 0.4 in Figures

4 and 5) also for moderate values of the correlation. Our results go therefore beyond

the conjectures of Jagannathan & Kocherlakota (1996) by stressing the dependence

of this inverse behavior also on other parameters of the model, like relative risk

aversion and volatility of the stock.

5. Conclusion

Given the rapidly increasing popularity of target date funds, the present paper

aims to find a rigorous theoretical foundation to justify their policy, and to find out

whether they can help the DC beneficiaries to effectively manage the investment risks

of their pension plans. Motivated by the qualitative results provided by Jagannathan

& Kocherlakota (1996) to verify the hypothesis young people shall invest more in

risky stocks, we come up with a more realistic continuous-time model setup to

examine the essence of TDFs. Compared to their paper, our model extends their

qualitative analysis to a more quantitative one, and allows us to examine the effect of

different time horizons. We designed two different contribution rules to describe the

contributions flowing to the 401(k) plans, both of which depend on salary. When the

contribution process is a function of salary only, say a constant fraction of the salary,

the resulting optimal strategy is in most cases a time-decreasing equity holding, i.e.

18

the common policy used in TDFs is justified. However, in some cases, e.g. when

the asset and salary are highly positively correlated, the optimal strategy could be

a time-increasing equity holding, which suggests that older beneficiaries shall invest

more in equity.

Appendix A. Computing the strategy 1. f(At , It) = (It) At

If we consider contribution rules of the form: f(x, y) = (y)x, where is exclu-

sively a function of y, it is possible to reduce the dimension of the HJB equation.

Assume indeed that the contribution rule f is of the type f(x, y) = (y)x, i.e. the

contributions are proportional to the fund value. Thinking of the basic case of the

Merton portfolio allocation problem, we could try to solve the HJB equation by

making an ansatz for the value function of the type v(t, x, y) = (t, y)U(x) to get

rid of the dependence on x. Recalling that in the power utility case

xU (x)

U(x)= ,

x2U (x)

U(x)= ( 1),

we get

t(t, y) = supA

{( + r)(t, y) + I(t, y)y(t, y)+ (A.1)

( 1)12

()2(t, y) + I(t, y)y(t, y)}

f(x, y)x

(t, y) I(t, y)y(t, y)1

2I(t, y)

2yy(t, y). (A.2)

This yields an independent two-dimensional PDE9.

Now, in the setting described in Example 3.1 we can directly take as the second

component of the controlled process instead of I, and the associated HJB equation

reads:

vt = supA

{xvx +

1

2(x)2vxx + xvxy

}+(r+y)xvx+(y)vy+

1

22vyy.

The ansatz approach works in this case, and enables us to reduce the previous

equation by one dimension. Deriving a reduced equation as in (A.2), it is then

9It is apparent that such an ansatz cannot work if the dependence on x of the function f is ofa different type (e.g. constant in x).

19

enough to find a function : R+ R R, C1,2, such that

t = supA

{(+ y)

1

222(1 )

}+ (r + y)+ ( y)y +

1

22yy,

(A.3)

(T, y) = 1, y R.

Taking a look at the supremum in this equation, it is easy to see that if the function

depends exponentially on y, the structure simplifies. This leads to the ansatz:

(t, y) = exp {(t)y + (t)} ,

for , : R+ R, C1functions such that (T ) = (T ) = 0. Computing thederivatives, we have

t(t, y) = (t, y)((t)y + (t))

y(t, y) = (t, y)(t)

yy(t, y) = (t, y)((t))2.

Substituting into the equation of the above expressions, we get for the optimal

strategy

=+ y(1 )

=

(1 )+

y(1 )

. (A.4)

Using now that the function is strictly positive, it is possible to factorize the

dependence on in Equation (A.3), and a polynomial of first degree in y is obtained.

For the equation to be satisfied, the coefficient of y as well as the constant term in the

equation above must be zero, and this condition yields the following two (integrable)

ODEs for and :{(t) = (t), t [0, T ](T ) = 0,{(t) = (+(t))

2

2(1) + r + (t) +122

2(t), t [0, T ](T ) = 0,

where the last expression derives from substituting (A.4) in the equation for .

20

If we recall that their solutions are given by:

(t) =

(1 e(Tt)

),

(t) =

[

2(1 )

( +

)2+ r + +

1

222

2

](T t)+[

1

( +

)

+ +

2

2

2

]e(Tt) 1

+[

2(1 )

22

(

)2 1

222

2

]e2(Tt) 1

2.

we can recover the optimal strategy, which is given in Equation (8). A standard

verification argument yields then that the value function is given by (t, y)U(x) and

the optimal strategy is as in Equation (A.4).

Appendix B. Computing the strategy 2. f(At , It) = f(It)

If we now take as controlled process the pair X = (A, c), we obtain, exactly as

in Appendix Appendix A, the HJB equation in terms of the parameters describing

the dynamics of the contribution process:

vt(t, x, y) = supA

{[x( + r) + y] vx(t, x, y) + C(t, y)vy(t, x, y)

+1

2(x)2vxx(t, x, y) +

1

2C(t, y)

2vyy(t, x, y) + C(t, y)xvxy(t, x, y)},

(B.1)

v(T, x, y) =U(x), (x, y).

To solve this optimal asset allocation problem, Chen, Mereu & Stelzer (2014) rely

on the properties of the value function, particularly homogeneity, to reduce the HJB

equation by one dimension and to make the optimization problem thus well solvable

through numerical methods. More concretely, the solution to the reduced problem

is

Theorem Appendix B.1 (Chen, Mereu & Stelzer (2014), Theorem 2.4). The valuefunction of the control problem (5) is given by

v(t, x, y) = yu

(t,x

y

), (t, x, y) [0, T ) (0,+) (0,+),

21

where u : [0, T ) R+ R is the unique viscosity solution with polynomial growthat infinity to the following equation

ut + uz + C(t) [u zuz] +1

22C(t)

[( 1)u 2( 1)zuz + z2uzz

]+ (B.2)

supA

{( + r)zuz +

1

2()2z2uzz + C(t)( 1)zuz C(t)z2uzz

}= 0,

Furthermore, the optimal strategy is given by t = h(t, At , ct

)where

h(t, x, y) :=

uzz(t,xy

)

uz(t,xy

)xy

C(t)

1 uzz(t,xy )uz(t,

xy

)xy

1

, if it belongs to A a.e.

22

Blake, D., Wright, D., & Zhang, Y. (2013). Target-driven investing: Optimal invest-

ment strategies in defined contribution pension plans under loss aversion. Journal

of Economic Dynamics and Control, 37 (1), 195209.

Boulier, J.-F., Huang, S. J., & Taillard, G. (2001). Optimal management under

stochastic interest rates: the case of a protected defined contribution pension

fund. Insurance: Mathematics and Economics, 28 (2), 173189.

Cairns, A. J., Blake, D., & Dowd, K. (2006). Stochastic lifestyling: Optimal dy-

namic asset allocation for defined contribution pension plans. Journal of Economic

Dynamics and Control, 30 (5), 843877.

Campbell, J. Y. & Viceira, L. M. (2002). Strategic asset allocation: portfolio choice

for long-term investors. Oxford University Press.

Chen, A., Mereu, C., & Stelzer, R. (2014). Optimal investment with time-varying

stochastic endowments. Submitted for publication, Available at SSRN 2458484.

Di Giacinto, M., Federico, S., Gozzi, F., & Vigna, E. (2014). Income drawdown op-

tion with minimum guarantee. European Journal of Operational Research, 234 (3),

610624.

Dybvig, P. H. & Liu, H. (2010). Lifetime consumption and investment: Retirement

and constrained borrowing. Journal of Economic Theory, 145 (3), 885907.

Gao, J. (2008). Stochastic optimal control of DC pension funds. Insurance: Math-

ematics and Economics, 42 (3), 11591164.

Jagannathan, R. & Kocherlakota, N. R. (1996). Why should older people invest less

in stocks than younger people. Federal Reserve Bank of Minneapolis Quarterly

Review, 20 (3), 1120.

Merton, R. (1969). Lifetime portfolio seclection under uncertainty: the continuous-

time case. Review of Economic Statistics, 51, 247257.

Merton, R. (1971). Optimum consumption and portfolio rules in a continuous time

model. Journal of Economic Theory, 3, 373413.

23

Morningstar Fund Research (2012). Target-date series research paper: 2012

industry survey. http://corporate.morningstar.com/us/documents/

MethodologyDocuments/MethodologyPapers/TargetDateFundSurvey_2012.

pdf.

Pham, H. (2009). Continuous-time stochastic control and optimization with financial

applications, volume 61. Springer.

Spitzer, J. J. & Singh, S. (2008). Shortfall risk of target-date funds during retirement.

Financial Services Review, 17, 143153.

Zhang, A., Korn, R., & Ewald, C.-O. (2007). Optimal management and inflation

protection for defined contribution pension plans. Blatter der DGVFM, 28 (2),

239258.

24

http://corporate.morningstar.com/us/documents/MethodologyDocuments/MethodologyPapers/TargetDateFundSurvey_2012.pdfhttp://corporate.morningstar.com/us/documents/MethodologyDocuments/MethodologyPapers/TargetDateFundSurvey_2012.pdfhttp://corporate.morningstar.com/us/documents/MethodologyDocuments/MethodologyPapers/TargetDateFundSurvey_2012.pdf

IntroductionModel setupContribution as a fraction of the fund valueContribution depending only on the incomeThe impact of correlationHuman capital

ConclusionComputing the strategy 1. f(At,It)= (It)At Computing the strategy 2. f(At, It)=f(It)